Properties

Label 2-165-1.1-c1-0-3
Degree $2$
Conductor $165$
Sign $-1$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s − 3-s + 3.82·4-s − 5-s + 2.41·6-s + 0.828·7-s − 4.41·8-s + 9-s + 2.41·10-s − 11-s − 3.82·12-s − 5.65·13-s − 1.99·14-s + 15-s + 2.99·16-s − 1.17·17-s − 2.41·18-s − 6.82·19-s − 3.82·20-s − 0.828·21-s + 2.41·22-s − 4·23-s + 4.41·24-s + 25-s + 13.6·26-s − 27-s + 3.17·28-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s + 0.313·7-s − 1.56·8-s + 0.333·9-s + 0.763·10-s − 0.301·11-s − 1.10·12-s − 1.56·13-s − 0.534·14-s + 0.258·15-s + 0.749·16-s − 0.284·17-s − 0.569·18-s − 1.56·19-s − 0.856·20-s − 0.180·21-s + 0.514·22-s − 0.834·23-s + 0.901·24-s + 0.200·25-s + 2.67·26-s − 0.192·27-s + 0.599·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + 2.41T + 2T^{2} \)
7 \( 1 - 0.828T + 7T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 - 4.82T + 41T^{2} \)
43 \( 1 + 8.82T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 2.34T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.89236308217832953012385481959, −11.08616747244285173223687672393, −10.24270735616098881995544842593, −9.366705082037274225159796771471, −8.145492917234143663137211812727, −7.44551346438669193859360856384, −6.31227099398064812462775450942, −4.60397357310367788518780208858, −2.19032666796141313339072907284, 0, 2.19032666796141313339072907284, 4.60397357310367788518780208858, 6.31227099398064812462775450942, 7.44551346438669193859360856384, 8.145492917234143663137211812727, 9.366705082037274225159796771471, 10.24270735616098881995544842593, 11.08616747244285173223687672393, 11.89236308217832953012385481959

Graph of the $Z$-function along the critical line